Physics 101: Lecture 15 Rotational KE, Angular Momentum

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Physics 101 Lecture 15Rotational KE, Angular
Momentum
Exam II

• Todays lecture will cover Textbook Chapter
  8.5-8.7
• Exam Friday 800AM
• Review session Thursday 900 AM

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Example falling beam
If the right rope breaks, what is the initial
angular acceleration of the beam?
2m
40kg
3m
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Example Rolling

• An hoop with mass M, radius R, and moment of
  inertia I MR2 rolls without slipping down a
  plane inclined at an angle ? 30o with respect
  to horizontal. What is its acceleration?
• Consider CM motion and rotation about the CM
  separately when solving this problem

I
R
M
?
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Rolling...

• Static friction f causes rolling. It is an
  unknown, so we must solve for it.
• First consider the free body diagram of the
  object and use
• In the x direction
• Now consider rotation
• about the CMand use ? I?

M
R
?
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Rolling...

• We have two equations
• We can combine these to eliminate f

I
A
R
M
?
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Rotational Energy

• It is moving so it is a type of Kinetic Energy
  (go back and rename the first)

Rotaional KE
Translational KE
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Example cylinder rolling

• Consider a cylinder with radius R and mass M,
  rolling w/o slipping down a ramp. Determine the
  ratio of the translational to rotational KE.

Friction causes object to roll, but if it rolls
w/o slipping friction does NO work! W F d cos q
d is zero for point in contact No dissipated
work, energy is conserved Need to include both
translation and rotation kinetic energy.
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Example cylinder rolling

• Consider a cylinder with radius R and mass M,
  rolling w/o slipping down a ramp. Determine the
  ratio of the translational to rotational KE.

Translational Rotational
Ratio
H
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Example cylinder rolling

• What is the velocity of the cylinder at the
  bottom of the ramp?

H
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Angular Momentum
Momentum Angular Momentum p mV L
I? conserved if ?Fext 0 conserved if ??ext
0 Vector Vector! units kg-m/s units
kg-m2/s
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Direction of Angular Momentum

• Right Hand Rule
• Wrap fingers of right hand around direction of
  rotation, thumb gives direction of angular
  momentum.

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Example Two Disks

• A disk of mass M and radius R rotates around the
  z axis with angular velocity ?i. A second
  identical disk, initially not rotating, is
  dropped on top of the first. There is friction
  between the disks, and eventually they rotate
  together with angular velocity ?f. Find ?f.

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Example Merry Go Round
Four students (mass 70kg) are riding on a
merry-go-round (solid disk of mass 90kg
rotating with angular velocity w3 rad/s.
Initially all four students are on the outer
edge. Suddenly 3 of students pull themselves to
within 0.25m of the center. What is the final
angular velocity of the merry-go-round?
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Example Merry Go Round
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Example Merry Go Round
What is the centripetal acceleration felt by each
of the students?
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Demo

• You are sitting on a freely rotating bar-stool
  with your arms stretched out and a weight in each
  hand. Your professor gives you a twist and you
  start rotating around a vertical axis though the
  center of the stool. You can assume that the
  bearing the stool turns on is frictionless, and
  that there is no net external torque present once
  you have started spinning.
• You now pull your arms and hands (and weights)
  close to your body.

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Demo

• What happens to the angular momentum as you pull
  in your arms?
• 1. it increases 2. it decreases 3. it stays the
  same

What happens to your angular velocity as you pull
in your arms? 1. it increases 2. it decreases
3. it stays the same
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Demo

• What happens to your kinetic energy as you pull
  in your arms?
• 1. it increases 2. it decreases 3. it stays the
  same

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Turning the bike wheel

• A student sits on a barstool holding a bike
  wheel. The wheel is initially spinning CCW in
  the horizontal plane (as viewed from above) L 25
  kg m2/s She now turns the bike wheel over. What
  happens?
• A. She starts to spin CCW.B. She starts to spin
  CW.C. Nothing

Start w/ angular momentum L pointing up from
wheel. When wheel is flipped, no more angular
momentum from it pointing up, so need to spin
person/stool to conserve L!
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Gyroscopic Motion

• Suppose you have a spinning gyroscope in the
  configuration shown below
• If the left support is removed, what will happen??

pivot
support
?
g
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Gyroscopic Motion...

• Suppose you have a spinning gyroscope in the
  configuration shown below
• If the left support is removed, what will happen?
• The gyroscope does not fall down!

pivot
?
g
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Gyroscopic Motion...
Bicycle wheel

• ... instead it precesses around its pivot axis !

pivot
?
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Summary

• St I a
• L I w
• Right Hand Rule gives direction
• If St 0, L is conserved